Optical image processing using minimum phase functions

Abstract

A method utilizes an optical image processing system. The method includes providing a measured magnitude of the Fourier transform of a complex transmission function of an object or optical image. The method further includes providing an estimated phase term of the Fourier transform of the complex transmission function. The method further includes multiplying the measured magnitude and the estimated phase term to generate an estimated Fourier transform of the complex transmission function. The method further includes calculating an inverse Fourier transform of the estimated Fourier transform, wherein the inverse Fourier transform is a spatial function. The method further includes calculating an estimated complex transmission function by applying at least one constraint to the inverse Fourier transform.

1. A method of utilizing an optical image processing system, the method comprising:
 (a) providing to the optical image processing system a measured magnitude of the Fourier transform of a complex transmission function of an object or optical image, wherein the complex transmission function equals or approximates a minimum-phase function or a maximum-phase function; 
 (b) providing to the optical image processing system an estimated phase term of the Fourier transform of the complex transmission function; 
 (c) using the optical image processing system for multiplying the measured magnitude and the estimated phase term to generate an estimated Fourier transform of the complex transmission function; 
 (d) calculating an inverse Fourier transform of the estimated Fourier transform, wherein the inverse Fourier transform is a spatial function; and 
 (e) calculating an estimated complex transmission function by applying at least one constraint indicative of a known property of the complex transmission function to the inverse Fourier transform by making the inverse Fourier transform conform to the known property. 
 
     
     
       2. The method of  claim 1 , further comprising using the optical image processing system for:
 (f) calculating a Fourier transform of the estimated complex transmission function; and 
 (g) calculating a calculated phase term of the Fourier transform of the complex transmission function. 
 
     
     
       3. The method of  claim 2 , further comprising:
 (h) using the calculated phase term of (g) as the estimated phase term of (c) and repeating (c)-(e). 
 
     
     
       4. The method of  claim 3 , wherein (c)-(h) are iteratively repeated until the estimated complex transmission function reaches convergence. 
     
     
       5. The method of  claim 4 , wherein convergence is reached when a difference between estimated complex transmission functions obtained after two consecutive iterations is less than a predetermined value. 
     
     
       6. The method of  claim 5 , wherein the predetermined value is 0.1% of the estimated complex transmission function of an iteration. 
     
     
       7. The method of  claim 3 , wherein (c)-(h) are iteratively repeated a predetermined number of times. 
     
     
       8. The method of  claim 1 , wherein the optical image processing system comprises a composite structure comprising an object of interest and an element near or next to the object of interest, the element having a first region with a first transmissivity and a second region with a second transmissivity less than the first transmissivity. 
     
     
       9. The method of  claim 8 , wherein providing the measured magnitude of the Fourier transform of a complex transmission function comprises:
 transmitting plane waves of light through the composite structure; 
 measuring a spatial frequency spectrum of the transmitted light; and 
 calculating the square root of the measured spatial frequency spectrum. 
 
     
     
       10. The method of  claim 9 , wherein the first region has a size larger than the wavelength of the light transmitted through the composite structure. 
     
     
       11. The method of  claim 10 , wherein the first region is at least a factor of two larger than the wavelength of the light transmitted through the composite structure. 
     
     
       12. The method of  claim 8 , wherein the second region comprises a metal layer on a glass layer, and the first region comprises a hole in the metal layer. 
     
     
       13. The method of  claim 8 , wherein providing the measured magnitude of the Fourier transform of the complex transmission function comprises providing a calculated square root of a previously-measured spatial frequency spectrum of light transmitted through the composite structure. 
     
     
       14. The method of  claim 1 , wherein providing the estimated phase term of the Fourier transform of the complex transmission function comprises providing an initial estimated phase term equal to a real or complex constant. 
     
     
       15. The method of  claim 14 , wherein the initial estimated phase term is a previously-stored function retrieved from a computer system. 
     
     
       16. The method of  claim 14 , wherein the initial estimated phase term is a phase calculated from a previous optical image. 
     
     
       17. The method of  claim 14 , wherein the initial estimated phase term is calculated from the measured magnitude using a logarithmic Hilbert transform. 
     
     
       18. The method of  claim 1 , wherein the complex transmission function equals or approximates a minimum-phase function and applying the at least one constraint to the inverse Fourier transform comprises setting the inverse Fourier transform to zero for selected portions of the complex transmission function. 
     
     
       19. A non-transitory tangible computer-readable storage medium having instructions stored thereon which cause a general-purpose computer of an optical image processing system to perform a method comprising:
 (a) providing to the optical image processing system a measured magnitude of the Fourier transform of a complex transmission function of an object or optical image, wherein the complex transmission function equals or approximates a minimum-phase function or a maximum-phase function; 
 (b) providing to the optical image processing system an estimated phase term of the Fourier transform of the complex transmission function; 
 (c) using the optical image processing system for multiplying the measured magnitude and the estimated phase term to generate an estimated Fourier transform of the complex transmission function; 
 (d) calculating an inverse Fourier transform of the estimated Fourier transform, wherein the inverse Fourier transform is a spatial function; and 
 (e) calculating an estimated complex transmission function by applying at least one constraint indicative of a known property of the complex transmission function to the inverse Fourier transform by making the inverse Fourier transform conform to the known property. 
 
     
     
       20. A computer system for processing an optical image comprising:
 means for estimating an estimated phase term of a Fourier transform of a complex transmission function of the optical image, wherein the complex transmission function equals or approximates a minimum-phase function or a maximum-phase function; 
 means for multiplying a measured magnitude of the Fourier transform of the complex transmission function and the estimated phase term to generate an estimated Fourier transform of the complex transmission function of the optical image; 
 means for calculating an inverse Fourier transform of the estimated Fourier transform, wherein the inverse Fourier transform is a spatial function; and 
 means for calculating an estimated complex transmission function of the optical image by applying at least one constraint indicative of a known property of the complex transmission function to the inverse Fourier transform by making the inverse Fourier transform conform to the known property; and 
 means for displaying the estimated complex transmission function of the optical image.